Algebraic Entropy of Generalized Shifts on Direct Products

For a set Γ, a function λ: Γ → Γ and a nontrivial abelian group K, the \emphgeneralized shift σ_λ: K ^Γ → K ^Γ is defined by (x _i ) _i∈Γ ? (x _λ(i)) _i∈Γ [3 Ayatollah Zadeh Shirazi , F. , Heidari Ardi , F. , Karami Kabir , N. ( 2008 ). A note on shift theory . Math. Pannon. 19 : 187 – 195 . [Google Scholar]]. In this article we compute the algebraic entropy of σ_λ; it is either zero or infinite, depending exclusively on the properties of λ. This solves two problems posed in [2 Akhavin , M. , Ayatollah Zadeh Shirazi , F. , Dikranjan , D. , Giordano Bruno , A. , Hosseini , A. ( 2009 ). Algebraic entropy of shift endomorphisms on abelian groups . Quaest. Math. 32 : 529 – 550 .[Crossref], [Web of Science ®] , [Google Scholar]].     

A Regularized Hybrid Steepest Descent Method for Variational Inclusions

This article is concerned with a generalization of the hybrid steepest descent method from variational inequalities to the multivalued case. This will be reached by replacing the multivalued operator by its Yosida approximate, which is always Lipschitz continuous. It is worth mentioning that the hybrid steepest descent method is an algorithmic solution to variational inequality problems over the fixed point set of certain nonexpansive mappings and has remarkable applicability to the constrained nonlinear inverse problems like image recovery and MIMO communication systems (see, e.g., [9 I. Yamada , M. Yukawa , and M. Yamagishi ( 2011 ). Minimizing the moreau envelope of nonsmooth convex functions over the fixed point set of certain quasi-nonexpansive mappings . In Fixed Point Algorithms for Inverse Problems in Science and Engineering ( H.H. Bauschke , R. Burachik , P.L. Combettes , V. Elser , D.R. Luke , and H. Wolkowicz , eds.), Springer-Verlag , New York , Chapter 17 , pp. 343 – 388 . [Google Scholar], 10 I. Yamada , Ogura , and N. Shirakawa ( 2002 ). A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems . In Inverse Problems, Image Analysis and Medical Imaging. Contemporary Mathematics ( Z. Nashed and O. Scherzer , eds.), American Mathematical Society , Providence , RI , Vol. 313 , pp. 269 – 305 . [Google Scholar]]).     

Non–existence of theta–shaped self–similarly shrinking networks moving by curvature

We prove that there are no networks homeomorphic to the Greek “Theta” letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees, such that under the motion by curvature they are self–similarly shrinking.

This fact completes the classification of the self–similarly shrinking networks in the plane with at most two triple junctions, see [5 Chen, X., Guo, J.-S. (2007). Self-similar solutions of a 2-D multiple-phase curvature flow. Phys. D. 229(1):22–34.[Crossref], [Web of Science ®] , [Google Scholar], 10 Hättenschweiler, J. (2007). Mean curvature flow of networks with triple junctions in the plane. Master’s thesis. ETH Zürich. [Google Scholar], 25 Schnürer, O. C., Azouani, A., Georgi, M., Hell, J., Nihar, J., Koeller, A., Marxen, T., Ritthaler, S., Sáez, M., Schulze, F., Smith, B. (2011). Evolution of convex lens–shaped networks under the curve shortening flow. Trans. Am. Math. Soc. 363(5):2265–2294.[Crossref], [Web of Science ®] , [Google Scholar], 2 Baldi, P., Haus, E., Mantegazza, C. (2016). Networks self-similarly moving by curvature with two triple junctions. Networks self-similarly moving by curvature with two triple junctions. 28(2017):323–338. [Google Scholar]].     

On the Characterization of f-Ideals

In this paper, we give the characterization of unmixed f-ideals of degree d ≥ 2 generalizing the results given in [1 Abbasi , G. Q. , Ahmad , S. , Anwar , I. , Baig , W. A. ( 2012 ). f-Ideals of degree 2. Algebra Colloquium 19 (Spec 1):921–926 . [Google Scholar]].     

Feebly Baer Rings and Modules

A right module M over a ring R is called feebly Baer if, whenever xa = 0 with x ∈ M and a ∈ R, there exists e² = e ∈ R such that xe = 0 and ea = a. The ring R is called feebly Baer if R_R is a feebly Baer module. These notions are motivated by the commutative analog discussed in a recent paper by Knox, Levy, McGovern, and Shapiro [6 Knox , M. L. , Levy , R. , McGovern , W. Wm. , Shapiro , J. ( 2009 ) Generalizations of complemented rings with applications to rings of functions. . J. Alg. Appl. 8 ( 1 ): 17 – 40 .[Crossref] , [Google Scholar]]. Basic properties of feebly Baer rings and modules are proved, and their connections with von Neumann regular rings are addressed.     

On Power-Associative Nilalgebras of Dimension N and Nilindex N-1

Whether or not a finite-dimensional, commutative, power-associative nilalgebra is solvable is a well-known open problem. In this paper, we describe commutative, power-associative nilalgebras of dimension n ≥ 6 and nilindex n ? 1 based on the condition that n ? 4 ≤ dim 𝔄³ ≤ n ? 3. This paper is a continuation of [10 Fernadez , J. C. G. , Garcia , C. I. , Montoya , M. L. R. ( 2013 ). On power-associative nilalgebras of nilindex and dimension n . Revista Colombiana de Matemáticas 47 : 1 – 11 . [Google Scholar]], where we describe commutative power-associative nilalgebras of dimension and nilindex n. We observe that the Jordan case was obtained by L. Elgueta and A. Suazo in [2 Elgueta , L. , Suazo , A. ( 2002 ). Jordan nilalgebras of nilindex n and dimension n + 1 . Comm. Algebra 30 : 5547 – 5561 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

Global Existence for a Kinetic Model of Chemotaxis via Dispersion and Strichartz Estimates

We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80's when experimental observations have shown they move by a series of ‘run and tumble’. The existence of solutions has been obtained in several papers Chalub et al. (2004 Chalub , F. A. C. C. , Markowich , P. A. , Perthame , B. , Schmeiser , C. ( 2004 ). Kinetic models for chemotaxis and their drift-diffusion limits . Monatsh. Math. 142 : 123 – 141 .[Crossref], [Web of Science ®] , [Google Scholar]), Hwang et al. (2005a Hwang , H. J. , Kang , K. , Stevens , A. ( 2005a ). Global solutions of nonlinear transport equations for chemosensitive movement . SIAM J. Math. Anal. 36 ( 4 ): 1177 – 1199 . [Google Scholar] b Hwang , H. J. , Kang , K. , Stevens , A. ( 2005b ). Drift-diffusion limits of kinetic models for chemotaxis: a generalization . Discrete Contin. Dyn. Syst. Ser. B 5 ( 2 ): 319 – 334 . [Google Scholar]) using direct and strong dispersive effects.

Here, we use the weak dispersion estimates of Castella and Perthame (1996 Castella , F. , Perthame , B. ( 1996 ). Estimations de Strichartz pour les équations de transport cinétique. [Strichartz’ estimates for kinetic transport equations.] C. R. Acad. Sci. Paris Sér. I 322 ( 6 ): 535 – 540 . [Google Scholar]) to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption.     

Unifying strongly clean power series rings

It is unknown whether a power series ring over a strongly clean ring is, itself, always strongly clean. Although a number of authors have shown that the above statement is true in certain special cases, the problem remains open, in general. In this article, we look at a class of strongly clean rings, which we call the optimally clean rings, over which power series are strongly clean. This condition is motivated by work in [10 Diesl, A. J., Dorsey, T. J., Garg, S., Khurana, D. (2012). A note on completeness and strongly clean rings, preprint. [Google Scholar]] and [11 Diesl, A. J., Dorsey, T. J., Iberkleid, W., LaFuente-Rodriguez, R., McGovern, W (2013). Strongly clean triangular matrices over abelian rings, preprint. [Google Scholar]]. We explore the properties of optimally clean rings and provide many examples, highlighting the role that this new class of rings plays in investigating the question of strongly clean power series.     

Conjugacy Classes of P-Cycles of Type D in Alternating Groups

We classify the conjugacy classes of p-cycles of type D in alternating groups. This finishes the open cases in [3 Andruskiewitsch , N. , Fantino , F. , Graña , M. , Vendramin , L. ( 2011 ). Finite-dimensional pointed Hopf algebras with alternating groups are trivial . Ann. Mat. Pura Appl 190 : 225 – 245 .[Web of Science ®] , [Google Scholar]]. Also we determine all the subracks of those conjugacy classes which are not of type D.     

Semilocal Convergence of Steffensen-Type Algorithms for Solving Nonlinear Equations

In this article, we provide a semilocal analysis for the Steffensen-type method (STTM) for solving nonlinear equations in a Banach space setting using recurrence relations. Numerical examples to validate our main results are also provided in this study to show that STTM is faster than other methods ([7 I. K. Argyros , J. Ezquerro , J. M. Gutiérrez , M. Hernández , and S. Hilout ( 2011 ). On the semilocal convergence of efficient Chebyshev-Secant-type methods . J. Comput. Appl. Math. 235 : 3195 – 3206 .[Crossref], [Web of Science ®] , [Google Scholar], 13 J. A. Ezquerro and M. A. Hernández ( 2009 ). An optimization of Chebyshev's method . J. Complexity 25 : 343 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]]) using similar convergence conditions.     

An Application of Darbo Fixed-Point Theorem to a Class of Functional Integral Equations

In this article, using Darbo's fixed-point theorem associated with the measure of noncompactness, we prove a theorem on the existence of the solutions of some nonlinear functional integral equations in the space of continuous functions on interval [0, a]. Our existence results include several existence results obtained earlier by Maleknejad et al. [7 K. Maleknejad , K. Nouri , and R. Mollapourasl ( 2009 ). Investigation on the existence of solutions for some nonlinear functional-integral equations . Nonlinear Anal. 71 : 1575 – 1578 .[Crossref], [Web of Science ®] , [Google Scholar]] and Özdemir et al. [8 ?. Özdemir , Ü. Çakan , and B. ?lhan ( 2013 ). On the existence of the solutions for some nonlinear Volterra integral equations . Abstr. Appl. Anal. Article ID 689234.  [Google Scholar]] as special cases under some weaker conditions. We give also some examples which show that our results are applicable.     

On an Ingelstam's Theorem

A well-known Ingelstam's Theorem asserts that every real Hilbert space A with an associative unital product satisfying ‖ xy‖ ≤ ‖ x‖ ‖ y‖ and ‖ 1‖ = 1 is isomorphic to the reals ?, or the complex numbers ?, or the quaternions ?. This note deals with a nonunital and nonassociative extension of the Ingelstam Theorem. So the assumptions about associativity and existence of unity are weakened to the existence of a nonzero central idempotent e such that ‖ ex‖ = ‖e‖ ‖ x‖ for all x, and that in A holds a determined kind of algebraic identity strictly weaker that alternativeness. We prove that, up to isomorphisms, there are only seven algebras satisfying these assumptions, even without the requirement of completeness. On the other hand, Section 3 presents another characterization of the obtained algebras with the flavor of one of the main theorems in Bhatt et al. (1998 Bhatt , S. J. , Karia , D. J. , Kulkarni , S. H. , Shimpi , M. E. ( 1998 ). A note on the Gelfand-Mazur theorem . Proc. Amer. Math. Soc. 126 ( 10 ): 2999 – 3005 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Construction of CPs-Stratified Algebras

The results of [7 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras. In: Tachikawa, H., Brenner, S. eds. Representations of Algebras and Related Topics, London Math. Society Lecture Note Series 168:200–224 . [Google Scholar]] and [2 Ágoston , I. , Dlab , V. , Lukács , E. ( 2011 ). Constructions of stratified algebras . Comm. Algebra 39 : 2545 – 2553 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] gave a recursive construction for all quasi-hereditary and standardly stratified algebras starting with local algebras and suitable bimodules. Using the notion of stratifying pairs of subcategories, introduced in [3 Ágoston , I. , Lukács , E. Stratifying pairs of subcategories for CPS-stratified algebras . To appear in Journal of Algebra and Its Applications , p. 11 . [Google Scholar]], we generalize these earlier results to construct recursively all CPS-stratified algebras.     

Dominance and Transmissions in Supertropical Valuation Theory

This article is a sequel of [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar]], where we defined supervaluations on a commutative semiring R and studied a dominance relation ? ≥ ψ between supervaluations ? and ψ on R, aiming at an enrichment of the algebraic tool box for use in tropical geometry.

A supervaluation ?: R → U is a multiplicative map from R to a supertropical semiring U, cf. [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar]], [7 Izhakian , Z. , Rowen , L. ( 2011 ). Supertropical matrix algebra . Israel J. Math. 182 ( 1 ): 383 – 424 .[Crossref], [Web of Science ®] , [Google Scholar]], [8 Izhakian , Z. , Rowen , L. ( 2010 ). Supertropical polynomials and resultants . J. Alg. 324 : 1860 – 1886 . (Preprint at arXiv:0902.2155.) [Crossref], [Web of Science ®] , [Google Scholar]], [5 Izhakian , Z. , Knebusch , M. , Rowen , L. Supertropical monoids: Basics and canonical factorization . Preprint at arXiv:1108.1880 . [Google Scholar]], [9 Maclane , S. ( 1998 ). Categories for the Working Mathemtician. , 4th ed. Springer Vereag . [Google Scholar]], with further properties, which mean that ? is a sort of refinement, or covering, of an m-valuation (= monoid valuation) v: R → M. In the most important case, that R is a ring, m-valuations constitute a mild generalization of valuations in the sense of Bourbaki [1 Bourbaki , N. Algèbre Commutative VI, §3 No. 1 . [Google Scholar]], while ? ≥ ψ means that ψ: R → V is a sort of coarsening of the supervaluation ?. If ?(R) generates the semiring U, then ? ≥ ψ iff there exists a “transmission” α: U → V with ψ = α ○ ?.

Transmissions are multiplicative maps with further properties, cf. [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar], Section 5]. Every semiring homomorphism α: U → V is a transmission, but there are others which lack additivity, and this causes a major difficulty. In the main body of the article we study surjective transmissions via equivalence relations on supertropical semirings. We put special emphasis on homomorphic equivalence relations. Even those are often much more complicated than congruences by ideals in usual commutative algebra.     

Scattering Theory Below Energy for a Class of Hartree Type Equations

We prove the global existence and scattering for the Hartree-type equation in H ^s (?³) the low regularity space s < 1. We follow the ideas in Colliander et al. (2004 Colliander , J. , Keel , M. , Staffilani , G. , Takaoka , H. , Tao , T. ( 2004 ). Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ?³ . Comm. Pure Appl. Math. 57 : 987 – 1014 .[Crossref], [Web of Science ®] , [Google Scholar]) to the Hartree-type nonlinearity, and also develop the theory of the classical multilinear operator modifying the L ^p estimate in Coifman and Meyer (1978 Coifman , R. , Meyer , Y. ( 1978 ). Au delá des opérateurs pseudo-differentiel . Astérisque, Société Mathématique de France 57 . [Google Scholar]).     

Examples of Commutative Right-Nilalgebras Over Small Fields

Correa et al. (2003 Correa , I. , Hentzel , I. R. , Labra , A. ( 2003 ). On nilpotence of commutative right nilalgebras of low dimension . Int. J. Math. Game Theory Algebra 13 ( 3 ): 199 – 202 . [Google Scholar]) proved that any commutative right-nilalgebra of nilindex 4 and dimension 4 is nilpotent in characteristic ≠ 2,3. They did not assume power-associativity. In this article we will further investigate these algebras without the assumption on the dimension and providing examples in those cases that are not covered in the classification concentrating mostly on algebras generated by one element.     

GRADINGS OF SIMPLE JORDAN ALGEBRAS AND THEIR RELATION TO THE GRADINGS OF SIMPLE ASSOCIATIVE ALGEBRAS

In this paper we describe all group gradings of the simple Jordan algebra of a non-degenerate symmetric form on a vector space over a field of characteristic different from 2. If we use the notion of the Clifford algebra, then we are able to recover some of the gradings on matrix algebras obtained in an entirely different way in [BSZ] Bahturin, Y., Seghal, S. and Zaicev, M. in press. Group Gradings of Associative Algebras. J. Algebra, [Web of Science ®] , [Google Scholar].

     

A Note on the Dimension Theory of ℤ n -Graded Rings

In this note we want to generalize some of the results in [1 Brewer , J. , Montgomery , P. , Rutter E. , Heinzer , W. ( 1973 ). Krull dimension of polynomial rings in “Conference on Commutative Algebra, Lawrence 1972.” . Springer Lecture Notes in Mathematics 311 : 26 – 45 .[Crossref] , [Google Scholar]] from polynomial rings in several indeterminates to arbitrary ? ⁿ -graded commutative rings. We will prove an analogue of Jaffard's Special Chain Theorem and a similar result for the height of a prime ideal 𝔭 over its graded core 𝔭*.